Heavy-tailed critical Galton--Watson processes with immigration

Consider a critical Galton--Watson branching process with immigration, where the offspring distribution belongs to the domain of attraction of a $(1 + \alpha)$-stable law with $\alpha \in (0,1)$, and the immigration distribution either (i) has finite mean, or (ii) belongs to the domain of attraction of a $\beta$-stable law with $\beta \in (\alpha, 1)$. We show that the tail of the stationary distribution is regularly varying. We analyze the stationary process, determine its tail process, and establish a stable central limit theorem for the partial sums. The norming sequence is different from the one corresponding to the tail of the stationary law. In particular, the extremal index of the process is $0$.